solution of a differential equation

If a differential equation is a rule about rates of change, then a solution is a function that actually OBEYS the rule. Picture the equation as setting an instruction at every point — 'here your slope must be this' — and a solution as a curve that, faithfully and everywhere, has exactly the slope the rule demands. Finding such a curve is the whole point of the subject.

Precisely, a solution of a differential equation on an interval is a function that, together with its derivatives, satisfies the equation at every point of that interval — turning it into a true identity. To check, you substitute the function and its derivatives into the equation and confirm both sides agree for all x in the interval. For example, y = e^(2x) is a solution of y' = 2y on the whole real line, because its derivative 2 e^(2x) equals 2 times e^(2x) everywhere. A solution always comes attached to an interval where it is defined and the equation holds — that interval matters.

Three honest cautions belong here. First, a solution is a function, valid on an interval, not just a single value. Second, most differential equations have many solutions — a whole family — unless extra conditions select one. Third, and most soberingly, the large majority of differential equations have NO solution expressible by elementary formulas; the neat closed-form answers of a textbook are a small, special collection. That is precisely why later we develop numerical methods (to approximate solutions) and qualitative methods (to describe them without a formula).